Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1990. Sectio Physicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 20)

Anatol Nowicki: Composite spacetime from twistors and its extensions

- 18 invariant under quaternionic orthogonal group OCd.;tK> covering the conformal six dimensional group 0C6,2): OCdjQ-D = U

aC4;D-D = 0C8;O n UCd,4) = UC6,2T C21) Tor details see ref: C73. Therefore one can look for the quaternionic extension of D=4 twistor formalism which can describe KM

0 Minkowski space. Now, let us consider the case Cii). First, we recall some basic properties of the quaternions tH, and recommend the references C8 3 on this subject. The quaternions [H constitute a four—dimensional real associative algebra with identity l=e

Q. Any quaternion q is given by the sum: q = o

eo + + cl2

e2 + °l3

e3 ^fj e K' P=0,l,2,3 C22) where the quaternionic units satisfy the following multiplication rule: e e. = -<5. . + <s. . , e. i,j,k = 1,2,3 C23) VJ i J i j k k * ^ * * ' Let us notice that the real numbers IR are naturally embedded in [H by identifying q

0e

Q = q

Q e K. For quaternions one can define a quaternionic conjugation Cso called principal involution) writting q = q

o - q

1e

1 - q

2e

2 - q

3e

3 C2da) and the norm |q|

2 = qq = q

2 + q

2 + q

2 C2db) Therefore, the algebra (H has the natural structure of the four—dimensional Euclidean space. Sometimes it is useful to identify a quaternion q with the ordered pair of complex numbers Cz

±,z

2) by q = z„ + e„z„ = Cq„ + q e ) + e„Cq_ + q,e,) C23) n 1 2 2 ^ a "3 3 2 2 ^ 1 3 We can see that quaternions are the natural extensions of the real numbers IR as well as complex ones C. Now, in analogy tp C4) for the given 2x2 quaternionic matrix Z we can associate Z—plane in fourdimensional quaternionic space D-t

4" -